And also these comments:
Because FN-1(·) is the distribution function of the highest value among a bidder’s N - 1 competitors, the bidding strategy displayed in Theorem 9.1 says that each bidder bids the expectation of the second highest bidder’s value conditional on his own value being highest. But, because the bidders use the same strictly increasing bidding function, having the highest value is equivalent to having the highest bid and so equivalent to winning the auction. So, we may say: In the unique symmetric equilibrium of a first-price, sealed-bid auction, each bidder bids the expectation of the second-highest bidder’s value conditional on winning the auction.
I don't understand the maths of this. I do sense it's logical you'd only bid what you expect to be the value the 2nd highest bidder has for the object, but I can't see how the equation of theorom 9.1 leads you to that conclusion. Could someone explain that to me? Please consider I'm not as proficient with mathematics as many of you are (otherwise I probably wouldn't need to make this thread), so I need to have this explained step-by-step, if you don't mind.